{"title":"Asymptotic behavior of the prediction error for stationary sequences","authors":"N. Babayan, M. Ginovyan","doi":"10.1214/23-ps21","DOIUrl":null,"url":null,"abstract":"One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\\le t\\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process $X(t)$. In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case - deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process $X(t)$ and the differential properties of its spectral density $f$, while for deterministic processes it is determined by the geometric properties of the spectrum of $X(t)$ and singularities of its spectral density $f$.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Surveys","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/23-ps21","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1
Abstract
One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process $X(t)$. In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case - deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process $X(t)$ and the differential properties of its spectral density $f$, while for deterministic processes it is determined by the geometric properties of the spectrum of $X(t)$ and singularities of its spectral density $f$.